C++ for Scientists - Technische Universität Dresden

2.11. REAL-WORLD EXAMPLE: MATRIX INVERSION 59
dense vector inline unit vector(unsigned k, unsigned n)
{
dense vector v(n, 0.0);
v[k]= 1;
return v;
}
As the function returns the unit vector we can just take it as argument of the triangular solver
res k= upper trisolve(A, unit vector(k, n));
For a dense matrix, MTL4 allows us to access a matrix column as column vector (instead of a
sub-matrix). Then we can assign the result vector directly without a loop.
Inv[irange(0, n)][k]= res k;
As short explanation, the bracket operator is implemented in a manner that integer indices
for rows and columns returns the matrix entry while ranges for rows and columns returns a
sub-matrix. Likewise, a range of rows and a single column gives you a column of the according
matrix — or part of this column. Vice versa, a row vector can be extracted from a matrix with
an integer as row index and a range for the columns.
This is an interesting example how to deal with the limitations as well as possibilities of C ++.
Other languages have ranges as part of their intrinsic notation, e.g. Python has a symbol ‘:’
for expressing ranges of indices. C ++ does not have this symbol but we can introduce a new
type — like MTL4’s irange — and define the behavior of operator[] for this type. This leads to
an extremely powerful mechanism!
Extending Operator Functionality
Since we cannot introduce new operators into C ++— not now (in 2010), not
in the next standard (C ++0x), maybe in that afterwards — we define new
types and give operators the desired behavior when applied to those types.
This technique allows us providing a very broad functionality with a limited
number of operators.
The operator semantics on user types shall be intuitive and must be consistent with the operator
priority (see example in § 2.3.7).
Back to our algorithm. We store the result of the solver in a vector and then we assign it to a
matrix column. In fact, we can assign the triangular solver’s result directly.
Inv[irange(0, n)][k]= upper trisolve(A, unit vector(k, n));
The range of all indices is predefined as iall:
Inv[iall][k]= upper trisolve(A, unit vector(k, n));
Next, we explore some mathematical back-ground. The inverse of an upper triangular matrix
is also upper triangular. Thus, we only need to compute the upper part of the result and set
the remainder to 0 — or the whole matrix to zero before computing the upper part. Of course,